## Steady flow energy equation :

For steady flow energy equation , Consider a control volume, in which mass ∆m1/∆t enters through inlet with parameters

U = internal energy

v = Velocity

V = specific volume ( V_{1 } = V_{2 })

P = pressure

Z= potential head

Similarly mass ∆m_{2}/∆t exits through outlet.

Therefore mass concentrationin control volume is

(∆m/∆t) _{control volume} = ∆m_{1}/∆t – ∆m_{2}/∆t

•(∆E/∆t)_{ in} = internal energy+ flow work + kinetic energy+ potential energy + heat

Therefore , (∆E/∆t)_{ in} = ∆m_{1}/∆t { U_{1 }+ P_{1}V_{1} + v_{1}^{2}/2 + gZ_{1 }} + ∆Q/∆t

At exit , (∆E/∆t)** _{out }** = ∆m

_{2}/∆t { U

_{2 }+ P

_{2}V

_{2 }+

**v**

_{2 2}/2 + gZ

_{2 }} + ∆W/∆t

(E/∆t)_{ control volume}= (∆E/∆t)_{in } – (∆E/∆t)_{out }

(E/∆t)_{ control volume}= ∆m_{1}/∆t { h_{1 } + v_{1}^{2}/2 + gZ_{1 }} + ∆Q/∆t – ∆m_{2}/∆t { h_{2 } + v_{2 2}/2 + gZ_{2} – ∆W/∆}t

**Assumptions :**

• (∆m/∆t) _{control volume} = 0

• (E/∆t)_{ control volume}= 0

Here , (∆m/∆t) _{control volume} = ∆m_{1}/∆t – ∆m_{2}/∆t = 0

So , ∆m_{1}/∆t = ∆m_{2}/∆t = m

Similarly,

(∆E/∆t)_{in} = (∆E/∆t)_{out }

Therefore ,

m{ h_{1} + v_{1}^{2}/2 + gZ_{1 }} + ∆Q/∆t = m{ h_{2} + v_{2 2}/2 + gZ_{2 }} + ∆W/∆t

this is **Steady flow energy equation** or **SFEE** .

## Bernoulli Equation :

from Steady flow energy equation

m{ h1 + v12/2 + gZ1 } + ∆Q/∆t = m{ h2 + v2 2/2 + gZ2 } + ∆W/∆t

Assume ,

(a)No heat transfer

(b) No work transfer

(c)No change in internal energy

(d)No change in density ( p = constant)

Then ,

m{ h1 + v12/2 + gZ1 } +~~ ∆Q/∆t ~~= m{ h2 + v2 2/2 + gZ2 } +~~ ∆W/∆t ~~

and

~~U~~_{1}_{ }+ P_{1}V_{1} + v_{1}^{2}/2 + gZ_{1} = ~~U~~_{2}_{ }+ P_{2}V_{2} + v_{2 2}/2 + gZ_{2}

As , Specific Volume = 1/ Density

i.e., V = 1/ p

Therefore

P_{1}/p + v_{1 } ^{2}/2 + gZ_{1 } = P_{2} /p+ v_{2 2}/2+ gZ_{2}

**P _{1}/pg + v_{1 2}/2g + Z_{1} = P_{2} /pg+ v_{2}^{2}/2g+ Z_{2}**

This is **Bernoulli Equation**

## Conclusion :

Hence both Steady flow energy equation and Bernoulli Equation are based on energy conservation and Bernoulli Equation is limited form of steady flow energy equation .

This is also known as the first law of thermodynamics for open system.